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Institute of Physics eBooks

Measurement theory in Deutsch’s algorithm based on the truth values

Koji Nagata, Do Ngoc Diep, Ahmed Farouk, Tadao Nakamura

Published Jul 1, 2022• 0 citations
Publisher

Abstract

We propose a new measurement theory, in qubits handling, based on the truth values, i.e., the truth T (1) for true and the falsity F (0) for false. The results of measurement are either 0 or 1. To implement Deutsch's algorithm, we need both observability and controllability of a quantum state. The new measurement theory can satisfy these two. Especially, we systematically describe our assertion based on more mathematical analysis using raw data in a thoughtful experiment.

DOI
10.1088/978-0-7503-4700-6ch20

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We propose a new measurement theory, in qubits handling, based on the truth values, i.e., the truth T (1) for true and the falsity F (0) for false. The results of measurement are either 0 or 1. To implement Deutsch's algorithm, we need both observability and controllability of a quantum state. The new measurement theory can satisfy these two. Especially, we systematically describe our assertion based on more mathematical analysis using raw data in a thoughtful experiment.
We consider the two quantum measurement theories for measuring a single Pauli observable. We assume also the existence of a classical probability space for the two measurement theories. We cannot avoid the Kochen-Specker (KS) contradiction when we measure the Pauli observable by using the projective measurement theory if we introduce a classical probability space. The results of measurement are either +1+1+1 or −1-1−1 (in ℏ/2\hbar/2ℏ/2 unit) when we consider a spin-1/2 system. The projective measurement theory does not accept a classical probability space when we measure the Pauli observable. We propose a new measurement theory based on the truth values, i.e., the truth T (1) for true and the falsity F (0) for false. The results of measurement are either +1+1+1 or 0 (in ℏ/2\hbar/2ℏ/2 unit). We avoid the KS contradiction when we measure the Pauli observable by using the new measurement theory if we introduce a classical probability space. The new measurement theory accepts a classical probability space when we measure the Pauli observable.
Recently, a new measurement theory based on truth values was proposed by Nagata and Nakamura [Int. J. Theor. Phys. 55 , 3616 (2016)], that is, a theory where the results of measurements are either 0 or 1. The standard measurement theory accepts a hidden variable model for a single Pauli observable. Hence, we can introduce a classical probability space for the measurement theory in this particular case. Additionally, we discuss in the present contribution the fact that projective measurement theories (the results of which are either +1 or −1) imply the Bell, Kochen, and Specker (BKS) paradox for a single Pauli observable. To justify our assertion, we present the BKS theorem in almost all the two-dimensional states by using a projective measurement theory. As an example, we present the BKS theorem in two-dimensions with white noise. Our discussion provides new insight into the quantum measurement problem by using this measurement theory based on the truth values .
Recently, a new measurement theory based on the truth values is proposed \cite{NN1}. The results of measurements are either 0 or 1. The measurement theory accepts a hidden variables model for a single Pauli observable. Therefore we can introduce a classical probability space for the measurement theory in this case. On the other hand, we discuss the fact that the projective measurement theory (the results of measurements are either +1+1+1 or −1-1−1) does not meet a hidden variables model for a single Pauli observable. Hence we cannot introduce a classical probability space for the projective measurement theory in this case. Our discussion provides new insight to formulate quantum measurement theory, by using the measurement theory based on the truth values.
What would be the consequences if there were fundamental limits to our ability to experimentally explore the world? In this work we seriously consider this question. We assume the existence of statements whose truth value is not experimentally accessible. That is, there is no way, not even in theory, to directly test if these statements are true or false. We further develop a theory in which experimentally accessible statements are a union of a fixed minimum number of inaccessible statements. For example, the value of truth of the statements "a" and "b" is not accessible, but the value of truth of the statement "a or b" is accessible. We do not directly assume probability theory, we exclusively define experimentally accessible and inaccessible statements and build on these notions using the rules of classical logic. We find that an interesting structure emerges. Developing this theory, we relax the logical structure to a probabilistic one, obtaining a theory rich in structure that we call "theory of inaccessible information". Surprisingly, the simplest model of theory of inaccessible information is the qubit in quantum mechanics. Along the path for the construction of this theory, we characterise and study a family of multiplicative information measures that we call "inaccessibility measures".
Measurements play an integral part in any scientific theory. Indeed, the observations revealing causal behaviour and symmetries in our world of experience lie at the heart of determining the truth or the falsity of scientific hypotheses. In this regard, quantum theory provides one of the most stringently tested and comprehensive description of the natural world. This thesis is an examination of the mathematical structures of quantum measurements with an emphasis on the measurement implementations of different operational devices: quantum observables, channels and instruments. The measurability of all the quantum properties a system bears is, in principle, guaranteed by quantum measurement theory. However, as physical processes, measurements are constrained by the laws of physics and it has been long known that certain conservation laws can inhibit the measurements of some of the quantum observables. In this work it is shown that such constraints stem from the fact that the measured observables inherit the symmetries of the measurement devices implementing them. A quantum measurement set-up can implement a multitude of operational devices as a function of the initial state of the apparatus. In other words, one can realize various maps – even incompatible ones – simply by controlling the apparatus’ state. Due its close resemblance with the classical analogue, this operation is often referred as quantum programming. In this thesis the capabilities and restrictions of quantum programming are explored using a measurement-theoretic approach. Moreover, quantitative methods are developed to address the mutual relations between the programming states of an arbitrary pair of observables or channels.